Calculus of Variations and Geometric Measure Theory

S. Bianchini - A. Brancolini

Estimates on Path Functionals over Wasserstein Spaces

created by brancolin on 26 Sep 2017


Published Paper

Inserted: 26 sep 2017

Journal: SIAM J. Math. Anal.
Volume: 42
Number: 3
Pages: 1179-1217
Year: 2010
Doi: 10.1137/100782693


In this paper we consider the class a functionals (introduced by Brancolini, Buttazzo, and Santambrogio) $\mathcal{G}_{r,p}(\gamma)$ defined on Lipschitz curves $\gamma$ valued in the $p$-Wasserstein space. The problem considered is the following: given a measure $\mu$, give conditions in order to assure the existence a curve $\gamma$ such that $\gamma(0) = \mu$, $\gamma(1) = \delta_{x_0}$, and $\mathcal{G}_{r,p}(\gamma) < +\infty$.

To this end, new estimates on $\mathcal{G}_{r,p}(\mu)$ are given and a notion of dimension of a measure (called path dimension) is introduced: the path dimension specifies the values of the parameters $(r,p)$ for which the answer to the previous reachability problem is positive. Finally, we compare the path dimension with other known dimensions.

Keywords: Path functionals, optimal transportation problems, branched transportation